On maximal stable orders on an inverse semigroup of finite rank with zero

Derech, V. D.

doi:10.1007/s11253-009-0132-1

Cited by 2 publications

(4 citation statements)

References 6 publications

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“…Moreover, in Theorem 2, we characterize the semilattice of idempotents of the finite inverse semigroup with zero all stable orders of which are exhausted by fundamental and antifundamental. The paper continues and develops (for the finite case) the results established in [7,8]. The principal result of the present paper is formulated in Theorem 1.…”

Section: Introductionmentioning

confidence: 75%

“…In [7,Theorem 1], it is shown that the function F : x R 1 ( x ) is a homomorphism from the semigroup S into the semigroup P ( I 1 ). Thus,…”

Section: Lemma 13 Let S Be a Finite Inverse Semigroup With Zero Ifmentioning

confidence: 99%

“…By virtue of Theorem 1 in [7], the function F : z R 1 ( z ) is a homomorphism from the semigroup S into the global supersemigroup P ( I 1 ). Hence, R 1 ( a a -1 a ) ⊆ R 1 ( a a -1 b ) or R 1 ( a ) ⊆ R 1 ( a a -1 b ).…”

Section: Lemma 16 Let S Be a Finite Inverse Semigroup With Zero Eachmentioning

confidence: 99%

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Structure of a finite inverse semigroup with zero every stable order on which is fundamental or antifundamental

Derech

2010

Ukr Math J

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Section: Introductionmentioning

confidence: 75%

“…In [7,Theorem 1], it is shown that the function F : x R 1 ( x ) is a homomorphism from the semigroup S into the semigroup P ( I 1 ). Thus,…”

Section: Lemma 13 Let S Be a Finite Inverse Semigroup With Zero Ifmentioning

confidence: 99%

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Structure of a finite inverse semigroup with zero every stable order on which is fundamental or antifundamental

Derech

2010

Ukr Math J

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“…Since the binary relation Ω is different from equality, there exist α, λ ∈T E such that ( , ) α λ ∈ Ω and α ≠ λ. In [9], it was proved in the general form that a function F : σ ‫ۋ‬ R 1 ( ) σ , where σ ∈T E , is a homomorphism from the semigroup T E into the supersemigroup P I ( ) 1 , i.e., the semigroup of all nonempty subsets of the ideal…”

Section: Lemma 11 Let E Be a Semilattice Of Finite Length Any Nonzermentioning

confidence: 99%

Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

Derech

2009

Ukr Math J

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We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental.The Munn semigroup (see [1]), i.e., the inverse semigroup of all isomorphisms between principal ideals of a semilattice with respect to the ordinary operation of composition of binary relations, plays a fundamental role in the theory of representations of inverse semigroups (see [2, p. 170]). This explains the importance of comprehensive investigation of these semigroups and their classification.In the present paper, we investigate the structure of a Munn semigroup of finite rank every order of which is fundamental or antifundamental (for definitions, see Sec. 1). The main result of this paper is Theorem 1. Main Definitions and TerminologyA semilattice E is called a semilattice of finite length if there exists a natural number n such that the length of any chain from E does not exceed n.Let E be a semilattice of finite length. It is obvious that it contains the least element, which we denote by 0. Let T E denote the Munn semigroup, i.e., the inverse semigroup of all isomorphisms between the principal ideals of the semilattice E with respect to the ordinary operation of composition of binary relations. It is clear that the transformation 0 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ is the least element of the semigroup T E . Denote the domain of definition and the range of values of the transformation f T E ∈ by dom ( ) f and im ( ) f , respectively. Let S be an arbitrary semigroup and let N 0 be the set of all nonnegative integers. A function rank : S → N 0 is called a rank function on the semigroup S if rank ( ) ab ≤ min ( ) rank a ( , rank ( ) b ) for any a b S , ∈ . The number rank ( )x is called the rank of the element x. Let S be an inverse semigroup whose lattice of idempotents has finite length. The function rank ( ) a = h aa -1 ( ) , where h aa -1 ( ) is the height of the idempotent aa -1 in the semilattice of idempotents of the semigroup S, is a rank function (see [3]). We say that an inverse semigroup is a semigroup of finite rank if the semilattice of its idempotents has finite length.Let f T E ∈ . If dom ( ) f = aE, then, by definition, rank ( ) f = rank ( ) a , where rank ( ) a is the height of the element a in the semilattice E. In [4], it was proved that rank : T E → N 0 is a rank function.An order relation τ on an arbitrary semigroup S is called fundamental (see [5] or [6, p. 289]) if the ordered semigroup ( ; ) S τ is O-isomorphic to a certain semigroup of partial transformations of a set ordered by in-

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On maximal stable orders on an inverse semigroup of finite rank with zero

Abstract: We consider maximal stable orders on semigroups that belong to a certain class of inverse semigroups of finite rank.

Cited by 2 publications

References 6 publications

Structure of a finite inverse semigroup with zero every stable order on which is fundamental or antifundamental

Structure of a finite inverse semigroup with zero every stable order on which is fundamental or antifundamental

Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

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